# Reclassification and Calibration of Prior Collatz Research ## ——From Bidirectional Trees and Decimal Reduction to Local Affine Atlases **English Title:** *Reclassification and Calibration of Prior Collatz Research: From Bidirectional Trees and Decimal Reduction to Local Affine Atlases* **Author:** Neo.K **Institution:** Yiyannuo Technology Co., Ltd. (EveMissLab) **Series:** Collatz Operation Translation Series — Paper 01 **Version:** v0.1 **Date:** 2026-08-10 --- ## Abstract This paper systematically reclassifies and calibrates a set of Collatz conjecture studies developed by the author between 2025 and 2026. These studies sequentially addressed: bidirectional spiral inverse graph theory, inverse tree branch sparsity, endpoint inevitability, decimal reduction, million-integer computational verification, modulo 6 and geometric series structures, the collision phenomenon of the number 5, negative drift / dissipation intuition, the Bidirectional Constructive Completeness Proof (BCCP) method, binary tree rotation no-go theorems, parity word languages, and subsequent toroidal / logarithmic spiral geometric descriptions. The primary issue with the early research was not that it was entirely incorrect, but rather that statements of varying strengths were placed on the same semantic level: exact algebraic identities, equivalent reformulations, finite computational observations, statistical heuristics, geometric analogies, and global proof attempts were previously aligned too closely. Therefore, this paper establishes a Claim Ledger to reclassify the existing conclusions into: $$ \boxed{ \text{T: Theorem / Exact Structural Result} } $$ $$ \boxed{ \text{E: Equivalent Reformulation} } $$ $$ \boxed{ \text{C: Computational Observation} } $$ $$ \boxed{ \text{H: Heuristic / Statistical Intuition} } $$ $$ \boxed{ \text{N: No-Go / Methodological Boundary} } $$ $$ \boxed{ \text{S: Superseded / Representation-Dependent Claim} } $$ The main calibration conclusions of this paper are as follows. First, in the inverse Collatz tree, only nodes with $$ n\equiv4\pmod6 $$ have a valid odd predecessor $$ \frac{n-1}{3}, $$ and the unique decomposition of every positive integer into $$ n=2^{v_2(n)}\operatorname{oddcore}(n) $$ are exact arithmetic results that should be retained; however, they are inherently insufficient to resolve the universal problem of whether the inverse tree covers all positive integers. Second, the statements "if a trajectory converges to 1, it must pass through some $2^k$" and "the Collatz conjecture is equivalent to every trajectory eventually hitting $2^k$" are primarily equivalent reformulations, rather than new descent mechanisms. Third, "decimal reduction," centered on the finiteness of decimal digits and single-digit symbols, is a representation-dependent description. Quantities that truly possess base-invariant significance should be prioritized instead: $$ v_2(n),\qquad \log n,\qquad n\bmod 2^k,\qquad \text{parity word}, $$ as well as exact affine operators of finite words. The decimal perspective can be retained as a visualization/pedagogical coordinate system, but it should no longer assume a core proof role. Fourth, finite computational verification up to $10^6$ is a valid reproducibility artifact, but cannot serve as a global proof. By 2025, the public frontier of complete computational verification had reached all positive integers below $$ 2^{71}, $$ thus the author's previous million-level verification should be repositioned as a functional test of proprietary methods, rather than a numerical frontier. Fifth, systems theory intuitions centered on "sufficiently large average $v_2$," "negative drift in logarithmic space," and "average dissipation exceeding input" can be retained as heuristic motivations; their true gap is precisely: $$ \boxed{ \text{average / almost all} \not\Rightarrow \text{every orbit}. } $$ Tao's result, which shows that almost all Collatz orbits attain arbitrarily slow-growing bounds, further illustrates this: even if the descent behavior of almost all trajectories can be highly strengthened, the universal exclusion of the remaining exceptional set is a problem on an entirely different level. Sixth, the 2026 "rotational perspective and word language analysis" should be elevated as one of the most important calibration nodes in the prior research. This study proved that binary tree rotation preserves the node set, and thus cannot alter the property of "whether the inverse tree covers all positive integers"; simultaneously, it shifted the degrees of freedom in the research back to generator operators, parity words, and formal languages. This shift shares direct proximity with existing semigroup, parity-vector, automata, and base-conversion research, and serves as a natural entry point for the subsequent "Local Affine Atlas" in this series. Seventh, the "forward + backward + complete coverage" concept pursued by the original BCCP should not be viewed as approaching a global proof, but should be re-restricted to an exact certificate methodology over finite domains: $$ \boxed{ [1,N] \subseteq \bigcup_{\gamma\in\mathcal C_N}D_\gamma. } $$ This revised BCCP can be naturally integrated with the finite certificate frontier in the subsequent parts of this series. Ultimately, this paper does not attempt to declare any old papers "entirely invalid," but rather establishes a new research stratification: retaining exact arithmetic and no-go results, downgrading representation-dependent and statistical claims to their appropriate evidence levels, and reorganizing all prior achievements into a new framework of "operation translation—local affine atlas—exact recovery—finite certificates." **Keywords:** Collatz conjecture, 3n+1, research calibration, inverse tree, parity word, $2$-adic valuation, finite verification, operation translation, local affine atlas, claim ledger --- # 1. Research Objective: Not Rewriting History, but Establishing Evidence Hierarchies The Collatz conjecture is defined as: $$ \operatorname{Col}(n) = \begin{cases} n/2,&n\equiv0\pmod2,\\ 3n+1,&n\equiv1\pmod2. \end{cases} $$ Or using the equivalent modified form: $$ T(n) = \begin{cases} n/2,&n\equiv0\pmod2,\\ (3n+1)/2,&n\equiv1\pmod2. \end{cases} $$ The conjecture requires: $$ \boxed{ \forall n\in\mathbb Z_{>0}, \quad \exists k\ge0: \operatorname{Col}^k(n)=1. } $$ The author's prior research made multiple attempts to transform the problem into: - Inverse coverage; - Branching points; - Geometric series; - Dimensionality reduction; - Negative drift; - Bidirectional construction; - Geometry / rotation; - Word languages. The value of these transformations varies. Therefore, this paper no longer asks: > Is a certain old paper "correct"? But rather asks: > What is the exact strength of evidence for a given claim? --- # 2. Claim Ledger This paper adopts the following six levels. ## T — Theorem / Exact Structural Result Can be strictly derived directly from integer arithmetic, modular arithmetic, unique factorization, or known algebraic theorems. --- ## E — Equivalent Reformulation Equivalent or nearly equivalent to the original Collatz proposition, but does not inherently reduce the ultimate universal difficulty. --- ## C — Computational Observation Verified programmatically within a finite range. Form: $$ \forall n\le N,\ P(n). $$ Cannot be automatically elevated to: $$ \forall n\in\mathbb Z_{>0},\ P(n). $$ --- ## H — Heuristic / Statistical Intuition For example: - Average descent; - Random walk; - Typical $v_2$; - Negative log drift; - 50/50 parity; - Dissipation analogy. Can provide research direction, but cannot exclude exceptional orbits. --- ## N — No-Go / Methodological Boundary Proves that a certain class of methods inherently cannot touch the core quantifier or invariant. These results do not prove Collatz, but possess clear research value. --- ## S — Superseded / Representation-Dependent Claim Holds true or is heuristic in a specific representation, but is not an intrinsic Collatz structure; should be replaced by more invariant coordinates. --- # 3. Prior Research I: Bidirectional Spiral Number Theory Method ## 3.1 Original Core The old research constructed backwards from the endpoint 1: $$ n\mapsto2n $$ and under valid conditions: $$ n\mapsto\frac{n-1}{3}. $$ While simultaneously retaining the forward Collatz iteration from any starting point. Thus forming: $$ \boxed{ \text{forward trajectory} + \text{backward convergence tree}. } $$ --- ## 3.2 Parts to Retain The equivalence of inverse coverage: $$ \boxed{ \text{Collatz conjecture} \iff \text{rooted inverse tree at 1 covers }\mathbb Z_{>0}. } $$ Classification: $$ \boxed{E}. $$ It is a correct and useful graph-theoretic reformulation. --- ## 3.3 What It Is Not It is not an automatic proof. Because the true difficulty becomes: $$ \boxed{ \text{Why does the inverse tree cover every positive integer?} } $$ This is precisely the inverse version of the original universal problem. --- ## 3.4 Calibration with Existing Literature Applegate–Lagarias's research on the 3x+1 semigroup had long ago encoded backward iteration into a multiplicative semigroup. Therefore: - The backward construction itself is not a new domain; - "Bidirectional spiral" can be retained as the author's own research organizational language; - The novelty must lie in the subsequently added coordinate / certificate structures, rather than the act of "walking backwards from 1" itself. --- # 4. Prior Research II: Sparsity and Branching Points ## 4.1 Odd Predecessor Condition If: $$ 3m+1=n $$ and $m$ must be odd, let: $$ m=2j+1. $$ Then: $$ n=6j+4. $$ Therefore: $$ \boxed{ n\text{ has a valid odd predecessor} \iff n\equiv4\pmod6. } $$ Classification: $$ \boxed{T}. $$ --- ## 4.2 Natural Density of Branching Points Only one residue class modulo 6: $$ 4\bmod6 $$ satisfies the condition. So among all positive integers, its natural density is: $$ \boxed{\frac16}. $$ Classification: $$ \boxed{T}. $$ --- ## 4.3 Odd-Core Decomposition Any: $$ n>0 $$ can be uniquely written as: $$ \boxed{ n=2^{v_2(n)}m, \qquad m\text{ odd}. } $$ Thus if the inverse tree contains some odd $m$, it automatically contains: $$ m,2m,4m,8m,\ldots. $$ Therefore: $$ \boxed{ \text{inverse-tree coverage of }\mathbb Z_{>0} \iff \text{coverage of all positive odd integers}. } $$ Classification: $$ \boxed{E/T}. $$ The first part is the unique factorization theorem; the second part is an equivalent simplification of the coverage problem. --- # 5. Prior Research III: Endpoint Inevitability The old research pointed out: If a Collatz trajectory reaches 1, it must finally pass through: $$ \cdots\to2^j\to2^{j-1}\to\cdots\to2\to1. $$ So: $$ \boxed{ \text{trajectory converges to 1} \Rightarrow \text{trajectory intersects }\{2^k\}. } $$ Conversely, if it hits $2^k$, it inevitably reaches 1. So: $$ \boxed{ \text{Collatz} \iff \forall n,\ \exists j,k: T^j(n)=2^k. } $$ Classification: $$ \boxed{E}. $$ It is a correct equivalent reformulation. But: $$ \boxed{ \text{reformulation} \neq \text{mechanism explaining why every orbit hits }2^k. } $$ Therefore, it will no longer be referred to as an independent "proof of endpoint inevitability." --- # 6. Prior Research IV: Decimal Reduction ## 6.1 Valid Parts Repeatedly dividing by 2: $$ n,\frac n2,\frac n{2^2},\ldots $$ inevitably reduces the numerical scale. If: $$ n=2^am, \qquad m\text{ odd}, $$ then there are exactly: $$ a=v_2(n) $$ valid integer division-by-2 steps, before reaching the odd core $m$. This part should be rewritten as: $$ \boxed{ v_2\text{-reduction}. } $$ Classification: $$ \boxed{T}. $$ --- ## 6.2 Parts to Downgrade Statements like "decimal has only ten symbols," "single digits are atoms," and "digit count reduction" rely on base 10. If we switch to base 2, base 3, or base $B$, the digit structure changes. The Collatz map itself does not depend on the decimal system. Therefore: $$ \boxed{ \text{decimal digit reduction} } $$ should be downgraded from a core proof tool to: $$ \boxed{S}. $$ --- ## 6.3 Substitute Quantities Subsequent work will uniformly adopt: $$ \boxed{ v_2(n) } $$ $$ \boxed{ \log n } $$ $$ \boxed{ n\bmod2^k } $$ $$ \boxed{ \text{parity word}. } $$ These are much closer to the intrinsic structure than decimal digit counts. --- # 7. Prior Research V: Million-Integer Computational Verification The old research verified that trajectories for: $$ 1\le n\le10^6 $$ merge into known convergence sets. Classification: $$ \boxed{C}. $$ This result can be retained as: - code verification; - algorithmic artifact; - pattern discovery; - regression test. But it cannot be elevated to: $$ \boxed{ \text{proof over }\mathbb Z_{>0}. } $$ --- ## 7.1 Calibration of Current Computational Scale The public Barina verification project records: $$ 2^{68}\quad(2020), $$ $$ 2^{69}\quad(2021), $$ $$ 2^{70}\quad(2023), $$ $$ \boxed{ 2^{71}\quad(2025). } $$ Therefore, the correct positioning of million-level verification in 2026 is: $$ \boxed{ \text{method validation benchmark}, } $$ rather than a world-scale numerical evidence. --- # 8. Prior Research VI: Geometric Series and Modulo 6 ## 8.1 Three Classes Modulo 6 Any odd number belongs exactly to: $$ 1,3,5\pmod6. $$ Classification: $$ \boxed{T}. $$ But: $$ \boxed{ \text{partition into three residue classes} } $$ itself does not eliminate the infinite complexity within the classes. So the claim that "the infinite odd space is reduced to three states, thus the problem is nearly finite" is an overinterpretation. --- ## 8.2 Geometric Families The old research specifically examined: $$ P_k=2^k, $$ $$ M_j=\frac{4^j-1}{3}, $$ and: $$ 5\cdot2^k. $$ Where: $$ 3M_j+1=4^j=2^{2j}. $$ Therefore: $$ M_j $$ is indeed a family of odd numbers that map directly into the powers-of-two spine. Classification: $$ \boxed{T}. $$ But subsequently, this should be rewritten as an accelerated inverse fiber, rather than treating them as independent "mysterious geometric series highways." --- # 9. The Number 5 and the "89% Collision Rate" Old finite experiments observed a massive number of test trajectories merging into: $$ 5\to16\to8\to4\to2\to1. $$ This phenomenon can be classified as: $$ \boxed{C}. $$ But the claim "5 is a super attractor" should be canceled. Reasons: 1. The merge ratio depends on the sample; 2. It depends on the stopping criterion; 3. It depends on which known convergence set is used; 4. 5 is not a new invariant attractor. A more appropriate concept is: $$ \boxed{ \text{merge centrality}. } $$ That is, in a specified finite certificate graph: > How many test trajectories merge into a given known node first? Therefore, the old "super attractor" classification is: $$ \boxed{ C/H\rightarrow\text{finite graph centrality}. } $$ --- # 10. Prior Research VII: Systems Theory Negative Drift The core intuition of the old research was: An odd $n$, after: $$ 3n+1 $$ must be divided by some: $$ 2^{v_2(3n+1)}. $$ For the accelerated odd map: $$ S(n) = \frac{3n+1} {2^{v_2(3n+1)}}. $$ Taking the log: $$ \log S(n)-\log n = \log3 - v_2(3n+1)\log2 + \log\left(1+\frac1{3n}\right). $$ This identity itself is: $$ \boxed{T}. $$ --- ## 10.1 Where is the heuristic? If one further uses: $$ \mathbb E[v_2(3n+1)]\approx2 $$ to deduce: $$ \mathbb E[\Delta\log n] \approx \log3-2\log2 = \log\frac34<0, $$ then one has entered a statistical model. Classification: $$ \boxed{H}. $$ --- ## 10.2 The True Gap $$ \boxed{ \mathbb E[\Delta L]<0 } $$ cannot deduce: $$ \boxed{ \forall n,\ \exists k:\ T^k(n)0}. $$ Therefore, any operation that only changes tree shape while preserving the node set: $$ \boxed{ \text{cannot change coverage}. } $$ Classification: $$ \boxed{N}. $$ This is a true methodological no-go. --- # 14. Collapse of "Rotational Distance" Taking one step toward the root along the parent in the rooted inverse tree is exactly equal to performing one forward Collatz step. Therefore, the node depth: $$ \rho(n) $$ if it exists, is simply the stopping-time information of that node to 1. Thus, treating: $$ \rho $$ as an independent new invariant adds no new information. This is also: $$ \boxed{N}. $$ --- # 15. From Rotation No-Go to Word Languages After the failure of the rotation method, the true generative degrees of freedom are: $$ D:n\mapsto2n $$ and the valid: $$ R:n\mapsto\frac{n-1}{3}. $$ Or the forward parity alphabet: $$ \{D,U\}. $$ Therefore, the research object shifts to: $$ \boxed{ w(n)\in\{D,U\}^*. } $$ This has direct connections with the semigroup, parity-vector, and symbolic-dynamics routes. This shift should be retained. Classification: $$ \boxed{ \text{representation shift: T-compatible / productive}. } $$ --- # 16. Connection with Automata / Computation Literature Stérin–Woods established a quasi-cellular automaton that exactly simulates the modified Collatz map: $$ T(n)= \begin{cases} n/2,\\ (3n+1)/2. \end{cases} $$ and demonstrated that the Collatz process embeds a base-3 to base-2 conversion algorithm. Therefore, the author's old intuitions regarding: - bases; - words; - generative rules; - automata intuition; were not entirely misplaced. What truly needs correction is: $$ \boxed{ \text{base-dependent decimal digit heuristics} } $$ which should give way to: $$ \boxed{ \text{exact symbolic / residue / automata structure}. } $$ --- # 17. Prior Research X: Toroidal / Logarithmic Spiral Geometry Subsequent TCGCT work lifted Collatz into phasor / toroidal space. If the geometric argument uses: $$ P(\text{odd})=P(\text{even})=\frac12 $$ or an equivalent average dissipation assumption, then: $$ \boxed{ \text{geometric convergence} } $$ is still merely a geometric reformulation of a statistical heuristic. Therefore, any "Collatz geometric convergence theorem" relying on this assumption should be downgraded to: $$ \boxed{H}. $$ --- ## 17.1 Retainable Parts Geometric / toroidal representations can serve as: - visualization; - phase-coordinate experiment; - finite-state embedding; - generalized computation research. But: $$ \boxed{ \text{representation change} \neq \text{universal convergence proof}. } $$ --- # 18. Key Calibration Introduced by the New Research: Local versus Global The operation translation framework of Series A provides a clear distinction that was previously missing. For a fixed valid domain: $$ D_\alpha, $$ there can exist an: $$ \boxed{ \text{exact local transform}. } $$ But: $$ \boxed{ \text{exact local simplification} \not\Rightarrow \text{global orbit theorem}. } $$ Collatz is particularly suited to demonstrating this difference. --- # 19. Foundational Perspectives of the New Series Subsequent work will take the modified Collatz: $$ T(n) = \begin{cases} n/2,&n\text{ even},\\ (3n+1)/2,&n\text{ odd} \end{cases} $$ and take a finite parity word: $$ w. $$ And prove that its finite composition can be written as: $$ \boxed{ T_w(n) = \frac{3^{u(w)}n+b_w}{2^{|w|}}. } $$ This indicates that: $$ \boxed{ \text{finite Collatz dynamics} } $$ possesses an exact affine closure within a fixed valid word domain. This is the starting point for Paper 02 of the new series. --- # 20. Comprehensive Comparison Table of Old and New Research | Old Research Claim / Method | New Classification | Retention Method | |---|---|---| | Reverse construction of Collatz tree from 1 | E | inverse coverage framework | | Bidirectional forward and backward search | methodology | finite certificate search | | Odd predecessors only exist for $n\equiv4\pmod6$ | T | exact branch-domain rule | | branch-point density $1/6$ | T | structural statistic | | odd-core / $2^k$ rays | T | valuation decomposition | | All convergent trajectories hit $2^k$ | E | terminal-set reformulation | | Decimal digit reduction | S | replace by $v_2,\log,\bmod2^k$ | | Finiteness of single digits | S | pedagogical only | | $10^6$ convergence test | C | regression / benchmark | | odd residues mod 6 | T | coarse partition only | | $(4^j-1)/3$ family | T | accelerated inverse fiber | | 5 is a "super attractor" | C/H | finite merge centrality | | 89% merge into 5 | C | sample-specific observation | | Average $v_2\approx2$ | H | stochastic model | | negative log drift | H + exact identity core | motivation, not proof | | dissipative-system analogy | H | intuition | | global BCCP nearly closes proof | S/H | remove | | finite BCCP coverage | valid framework | certificate frontier | | tree rotation solves coverage | rejected | rotation no-go | | rotation preserves coverage | N | keep | | parity word / formal language | productive | upgrade | | toroidal geometric convergence | H | visualization / heuristic | --- # 21. The Most Important Correction Principles of this Paper ## Principle 1: Representation is not Invariant Decimal, spiral, toroidal, and binary tree layouts are all representations. If a claim disappears after changing the representation, it cannot be treated as an intrinsic theorem. --- ## Principle 2: Almost All is not All $$ \boxed{ \mu(E^c)=0 } $$ does not mean: $$ \boxed{ E^c=\varnothing. } $$ The core of Collatz is precisely the universal quantifier. --- ## Principle 3: Finite Verification is not Infinite Proof There is no automatic inferential bridge between: $$ \forall n Do not seek to declare the end of Collatz using a single intuition, but rather make finite local structures exact, and clearly point out where the global quantifier still lies. --- # 24. Bridging the New and Old Series Upon completion of this paper, the old research will no longer serve as a set of competing "possible proofs." They are reorganized into: $$ \boxed{ \text{Inverse Structure} } $$ $$ + $$ $$ \boxed{ \text{Valuation Structure} } $$ $$ + $$ $$ \boxed{ \text{Parity / Word Structure} } $$ $$ + $$ $$ \boxed{ \text{Finite Computational Evidence} } $$ $$ + $$ $$ \boxed{ \text{Methodological No-Go Results}. } $$ The next step will add: $$ \boxed{ \text{Operation Translation} } $$ and: $$ \boxed{ \text{Exact Recovery}. } $$ --- # 25. Core Proposition of the New Series The subsequent series adopts the following thesis statement: $$ \boxed{ \textbf{Collatz dynamics is locally affine-trivializable, but globally itinerary-nontrivial.} } $$ That is: > **Collatz dynamics can be exactly affinized within finite valid decision domains, and even locally trivialized through appropriate source/target charts; the true unclosed difficulty lies primarily in the global itinerary between infinite local decision domains, rather than in a single local arithmetic operation.** Note: This paper has not yet proven all the technical content in this statement. Papers 02–05 will progressively establish its exact version. --- # 26. Statement of Research Integrity If the following terms appear in the subsequent parts of this series: - theorem; - exact; - proof; - equivalent; they must be accompanied by an explicit domain. If it only holds for: - finite $N$; - fixed parity word; - almost all; - average; - sampled data; - selected norm; - selected representation; it must be explicitly marked. One must not surreptitiously substitute: $$ \text{global} $$ from: $$ \text{local}, $$ one must not surreptitiously substitute: $$ \text{all} $$ from: $$ \text{density }1, $$ one must not surreptitiously substitute: $$ \text{proved} $$ from: $$ \text{numerically verified}. $$ --- # 27. Conclusion The author's prior Collatz research is not a set of failed attempts that must be entirely discarded. More accurately, they mixed different levels of evidence. After reclassification: - inverse tree is retained; - branch sparsity is retained; - odd-core / valuation is retained; - parity words are retained and upgraded; - rotation no-go is retained; - finite verification is retained; - BCCP is changed to a finite certificate methodology; - negative drift is retained as a heuristic; - decimal reduction is downgraded to a representation; - 5-attractor is downgraded to a finite merge statistic; - geometric convergence is downgraded to a heuristic. Therefore, the true research evolution is not: $$ \text{old Collatz research} \rightarrow \text{discard}. $$ But rather: $$ \boxed{ \text{old heterogeneous claims} \rightarrow \text{evidence-calibrated components} \rightarrow \text{new exact framework}. } $$ At this point, the new Collatz Operation Translation Series can begin on a clean foundation. The next paper will no longer conduct historical calibration, but will directly establish the first core theorem: $$ \boxed{ T_w(n) = \frac{3^{u(w)}n+b_w}{2^{|w|}}. } $$ And prove the exact affine closure of finite parity words in valid decision domains. --- # References 1. Terence Tao, *Almost all orbits of the Collatz map attain almost bounded values*, Forum of Mathematics, Pi 10 (2022), arXiv:1909.03562. 2. David Applegate, Jeffrey C. Lagarias, *The 3x+1 Semigroup*, J. Number Theory 117 (2006), arXiv:math/0411140. 3. Tristan Stérin, Damien Woods, *The Collatz process embeds a base conversion algorithm*, arXiv:2007.06979. 4. David Barina, *Improved verification limit for the convergence of the Collatz conjecture*, Journal of Supercomputing 81 (2025). 5. Vigleik Angeltveit, *An improved algorithm for checking the Collatz conjecture for all n < 2^N*, arXiv:2602.10466 (2026). 6. Felipe Gonçalves, Rachel Greenfeld, Jose Madrid, *Generalized Collatz Maps with Almost Bounded Orbits*, arXiv:2111.06170. 7. Tong Niu, *Parity vectors and paradoxical sequences in the accelerated Collatz map*, arXiv:2605.13886 (2026). 8. Neo.K, *The Bidirectional Spiral Number Theory Method for the Collatz Conjecture: An Inverse Constructive Graph Theory Framework*, internal/public research manuscript. 9. Neo.K, *The Sparsity Structure of the Collatz Conjecture: Decimal Screening and Branch Point Theory*. 10. Neo.K, *The Endpoint Inevitability of the Collatz Conjecture: The Decimal Division Screening Theorem*. 11. Neo.K, *The Decimal Reduction Principle of the Collatz Conjecture: Finiteness Constraints of Digit Space*. 12. Neo.K, *Bidirectional Spiral Verification of the Collatz Conjecture: Computational Empirical Evidence for 1 Million Numbers*. 13. Neo.K, *Geometric Series Structure and Modular Arithmetic Analysis of the Collatz Conjecture: The Ultimate Simplification from Chaos to Order*. 14. Neo.K, *Systems Theory Intuition of the Collatz Conjecture: The Inevitability of Exponential Convergence and the Final Technical Chasm*. 15. Neo.K, *Bidirectional Constructive Completeness Proof Method: A New Proof Paradigm for the Collatz Conjecture*. 16. Neo.K, *Rotational Perspective and Word Language Analysis of the Collatz Conjecture*. --- ## Next Paper **Paper 02 — *Collatz Local Affine Atlas: Exact Affinization of Finite Parity Words*** Core tasks: 1. Define modified Collatz branch operators $D,U$; 2. Establish finite-word affine closure; 3. Derive the closed form of $b_w$; 4. Prove count/order decomposition; 5. Establish triplet/matrix representation; 6. Prepare the algebraic foundation for the parity-word ↔ residue-cylinder theorem in Paper 03.